IN RECENT years, there have been significant efforts to develop

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Effective Bandwidth of Multimedia Traffic in Packet Wireless CDMA

Networks with LMMSE Receivers: A Cross-Layer Perspective

Fei Yu and Vikram Krishnamurthy

Abstract— We propose a novel concept of cross-layer effective bandwidth that characterizes the unified resource usage taking into account both physical layer linear minimum mean square error (LMMSE) receivers and statistical characteristics of the packet traffic in code division multiple access (CDMA) networks. Based on the concept of cross-layer effective bandwidth, we develop an optimal connection admission control (CAC) scheme for variable bit rate packet traffic with QoS constraints at both physical and network layers. By introducing a small signal-to-interference ratio (SIR) outage probability using the concept of cross-layer effective bandwidth, the capacity of CDMA networks in the proposed CAC scheme can be increased significantly compared to some existing schemes. The effectiveness of the proposed approaches is demonstrated by numerical examples. Index Terms— Code division multiple access, effective band-width, cross-layer design, admission control.

I. INTRODUCTION

I

N RECENT years, there have been significant efforts to de-velop sophisticated code division multiple access (CDMA) physical layer communication techniques, e.g., multiuser re-ceivers that mitigate not only the background noise but also the interference from other users of the multiaccess channel. In particular, the linear minimum-mean square error (LMMSE) receiver [1], [2] has received considerable attention. A notion of effective bandwidth is proposed in [1] to characterize the resource needed for a connection to meet the signal-to-interference ratio (SIR) requirement.

Although much work has been done in the analysis of CDMA networks, most of previous work concentrates on physical layer and does not consider the stochastically varying traffic load. Consequently, the obtained results may only ap-plicable to a “worst case” scenario withconstant bit rate traf-fic. However, future wireless systems are expected to support variable bit ratepacket traffic. In order to properly analyze a CDMA system, it is important to consider the characteristics of the carried traffic at both physical and network levels.

The concept of effective bandwidth of a traffic source at network level has been well developed [3], where the physical layer is assumed to be a wireline network. The network layer effective bandwidth may not be applicable to CDMA networks with LMMSE receivers as wireless fading channels are not considered in [3].

Manuscript received April 15, 2004; revised September 9, 2004; accepted January 14, 2005. The associate editor coordinating the review of this letter and approving it for publication was R. Fantacci. This work is supported by the Natural Science and Engineering Research Council of Canada.

The authors are with the Department of Electrical and Computer Engineer-ing, University of British Columbia, 2356 Main Mall, Vancouver, BC, Canada V6T 1Z4 (e-mail:{feiy, vikramk}@ece.ubc.ca).

Digital Object Identifier 10.1109/TWC.2006.03009.

Since the concept of effective bandwidth is useful at both physical and network levels, why not consider them jointly? In this letter, we propose a novel notion of cross-layer effective bandwidth, by which we can have a unified measure of resource usage taking into account the QoS requirements at both physical and network levels. The contributions of this work are two-fold:

1) The concept of cross-layer effective bandwidth is new. Unlike the previous work [1], [3] that considers only one layer (e.g., only physical layer or only network layer), our approach considers both LMMSE physical layer and statistical characteristics of the packet traffic at network layer.

2) Based on the concept of cross-layer effective bandwidth, we propose an optimal connection admission control (CAC) scheme for variable bit rate packet traffic with QoS constraints at both physical and network layers. Instead of guaranteeing the SIR at all time instants as in [4], [5], we guarantee the SIR outage probability. With a small SIR outage probability, the proposed CAC scheme can dramatically improve the net-work utilization. Unlike the CAC schemes based on Markov modulated Poisson process (MMPP) traffic models in [6], a variety of traffic models can be used in the proposed CAC scheme with the help of cross-layer effective bandwidth.

Authors in [7] propose to use the effective bandwidth con-cept in CDMA networks. However, the evaluation of CDMA capacity in [7] is heuristic as there is no mention of the receiver structure. Therefore, the effective bandwidth concept in [7] may only characterize the stochastically varying traffic but no fading channels. The predictive temporal structure of the multi-access interference (MAI) is investigated in [8] taking into account both channel fading and traffic burstiness. However, effective bandwidth is not considered in [8].

In numerical examples, we show that the admissible region obtained from the cross-layer effective bandwidth approach has good approximation to the simulations. In addition, the capacity of CDMA networks can be increased significantly in the proposed CAC scheme compared to the schemes in [4], [5], [7].

The rest of the letter is organized as follows. Section 2 describes the CDMA physical layer and network layer effective bandwidth. Section 3 presents the definition of cross-layer effective bandwidth. Section 4 presents the optimal CAC scheme. Some numerical examples are given in Section 5. Finally, we conclude this study in Section 6.

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II. MODELDESCRIPTION

A. Fading Channel and Linear Multiuser Detector Physical Layer Model

As in [1], [4], [5], we consider a synchronous CDMA system with spreading gain N andK users.1 The signature

sequences of all users are independent and randomly chosen. Due to multi-path fading, each user appears asL resolvable paths at the receiver. The pathl of userk is characterized by its estimated average channel gain¯hkland its estimation error

varianceξk2. The channel model is conditioned on the slow fad-ing (free space path loss and shadow fadfad-ing). The attenuated transmitted power from userkisPk =ZkPkt, k= 1,2, . . . , K,

whereZk is the path loss and Pkt is the transmitted power.

Linear minimum mean square error (LMMSE) detectors are used at the receiver. In a large system (bothNandKare large) with background noiseσ2, the SIR for the LMMSE receiver of a user (say, the first one) can be expressed approximately as [2] SIR1 = (P1Ll=1|h¯1l|2η)/(1 +P1ξ21η), where η is

the unique fixed point in (0,∞) that satisfies η = [σ2 +

(1/N)Kk=2((L 1)I(ξ2k, η) +I(

L

l=1|h¯kl|2 +ξk2, η))]1

andI(ν, η) =ν/(1 +νη).Assume that there areJ classes of traffic in the system. An important physical layer performance measure of classj users is SIRj, which should be kept above

the target valueωj. In [5], it is shown that a minimum received

power solution exists such that all users in the system meet their target SIRs if and only ifωj <|¯hij|2/ξji

2 and J j=1 nj i=1 Rij Υi j N <1, (1) where|¯hi

j|2is the average channel gain of theith classjuser,

|¯hij|2=

L l=1|¯h

i

jl|2,nj is the number of classj users,Rij is

the number of signature sequences assigned to theith user of classjto make it transmit atRijtimes the basic rate (obtained

using the lowest spreading gainN) and

Υi j = (L−1)ωj ξi j 2 |¯hi j|2 + ωj 1 + ξij2 |h¯ij|2 1 +ωj . (2)

The minimum received power solution for the first class 1 user is [5] P1 = ω1σ2/|¯h11|2(1 ω1ξ112/|¯h11|2)(1 (1/N)Jj=1 nj i=1R i jΥ i

j). The capacity of the system is

restricted by the power control feasibility condition (1). The SIR outage probability in CDMA networks with LMMSE receivers can be expressed as

Pout =P ⎧ ⎨ ⎩ J j=1 nj i=1 Rij Υi j N 1 ⎫ ⎬ ⎭. (3)

B. Network Layer Effective Bandwidth

Consider a bufferless communication multiplexer with a single output. There are J classes of input traffic with nj

1The proposed cross-layer effective bandwidth concept is also applicable to an asynchronous CDMA system. In [9], it is shown that an asynchronous CDMA system with K users can be viewed as a synchronous CDMA system withMK users, where M is the frame length. For simplicity of the presentation, we only consider synchronous systems in this letter.

connections of class j traffic. The aggregate input traffic is

Y =Jj=1ni=1j Yji, whereYji is theith class j traffic.

As-sume that the output capacity isC. The congestion probability of this system is Pcong =P ⎧ ⎨ ⎩ J j=1 nj i=1 Yji≥C ⎫ ⎬ ⎭. (4)

Given the statistical characteristics of traffic sources and their congestion probability requirements, the actual bandwidth that a connection requires lies between its mean rate and its peak rate. This bandwidth is generally referred to as the effective bandwidth of the traffic source. Assume that Yj[0, t] is the

amount of work that arrives from a class j source in the time interval[0, t], andYj[0, t]has stationary increments. The

definition of network layer effective bandwidth of class j

traffic is [3] αj(s, t) = 1 stlogE esYj[0,t] , s, t∈IR+, (5)

where E is the expectation, s and t are system parame-ters defined by the characteristics of the source, its QoS requirements, and the link capacity. The time parameter t

(measured in, e.g., milliseconds) indicates the time scales that are important for buffer overflow in buffer models: A small value of t indicates that fast time scales are responsible for buffer overflow, whereas a large value oftindicates that slow time scales are responsible for buffer overflow. In bufferless models, the instantaneous arrival rates of individual traffic sources contribute to packet loss, corresponding to the choice

αj(s) = limt0αj(s/t, t). The space parameters(measured

in, e.g., bits1) indicates the degree of statistical multiplexing and depends on the size of the peak rates of the multiplexed sources relative to the link capacity. Particularly, for the link capacity much larger than the peak rates of the multiplexed sources,stends to zero andαj(s, t)approaches the mean rate

of the source, corresponding to a large degree of statistical multiplexing, while for the link capacity not much larger than the peak rates of the sources,sis large andαj(s, t)approaches

the peak rate of the source, corresponding to a low degree of statistical multiplexing.

III. CROSS-LAYEREFFECTIVEBANDWIDTH A. Definition of Cross-Layer Effective Bandwidth

Comparing (3) with (4), we can see the similarity be-tween CDMA networks with LMMSE receivers and wireline networks. From a mathematical point of view, there is a scalar Υi

j/N in (3). It is very interesting to observe that the

scalor Υi

j/N contains all the information about the physical

layer LMMSE receivers. Since the definition of network layer effective bandwidth (5) is useful in deriving the congestion probability (4), we can develop a concept of cross-layer effective bandwidth to derive the SIR outage probability (3). This motivates us to define the cross-layer effective bandwidth of a traffic source as follows:

Definition 3.1: LetLdenote the number of resolvable paths that each user appears at the LMMSE receiver,|h¯i

j|2denote the

estimated average channel gain andξji

2

denote the estimation error variance of theith classj connections with a SIR target

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value ωj in a CDMA system with spreading gain N. Let

Ri

j[0, t] denote the amount of work generated from the ith

connection of class j in the time interval [0, t], and Ri j[0, t]

is assumed to have stationary increments. The cross-layer effective bandwidth of this connection is

whereΥi

j is defined in (2).

Remark:The definition above incorporates both physical layer LMMSE receivers and network layer packet traffic informa-tion. Specifically,Ri

j[0, t] contains the statistical

characteris-tics of the packet traffic, whereas the information about the physical layer LMMSE receivers is abstracted by Υij/N in

(6).L, N, h, ξ, ω are systems parameters defined by physical

layer LMMSE receivers. System parameterss, t are defined similarly as those of network layer effective bandwidth (5). The physical interpretation of αj(L, N, h, ξ, ω, s, t) is that

using the cross-layer effective bandwidth concept we can have a unified measure of resource usage given the physical layer wireless fading channels in CDMA systems with LMMSE receivers, statistical characteristics of the packet traffic and QoS requirements at both physical and network layers.

B. Derivation of SIR Outage Probability Using Cross-Layer Effective Bandwidth

We use SIR outage probability as a QoS measure in wireless packet CDMA networks. Instead of guaranteeing the SIR at all time instants, we guarantee the SIR outage probability. This formulation is motivated by the design of packet wireline networks. It is well known [10] that allocating all connections with their peak rates guarantees no packet loss, but results in the lowest utilization. Therefore, most bandwidth allocation schemes in wireline networks allow a small packet loss probability to increase the network utilization [10]. Similarly, since most applications in wireless networks can tolerate small probability of SIR outage, we can use SIR outage probability as a QoS measure and keep it below a target valueζ. The SIR outage probability can be estimated by the following well-knownChernoff bound[11] approximation

Pout=P ⎧ ⎨ ⎩ J j=1 nj i=1 Rij Υi j N 1 ⎫ ⎬ ⎭≈eΛ(1), (7) where Λ(v) = infs[sα−sv], α = J j=1 nj i=1αij, and αi

j is the scaled logarithmic moment generating

func-tion of the instantaneous work load of the ith class j

connection in a packet bufferless CDMA system. αij =

limt→0αij(L, h, ξ, ω, s/t, t) = (1/s) logE

esRijΥij/N

. The

constraint Pout ζ will be satisfied if the vector x =

(n1, n2, . . . , nJ)lies within the admissible set

X = ⎧ ⎨ ⎩x∈ZJ+: exp ⎧ ⎨ ⎩infs ⎡ ⎣s ⎛ ⎝J j=1 nj i=1 αij−1 ⎞ ⎠ ⎤ ⎦ ⎫ ⎬ ⎭≤ζ ⎫ ⎬ ⎭. (8) The Chernoff bound (7) can be further refined [12] by adding a prefactor.Pout≈eΛ(1)/s∗2π∂2(s∗α)/∂s2,wheres∗attains the infimum in (8). The admissible set using the improved bound becomes

IV. OPTIMALCONNECTIONADMISSIONCONTROL IN

PACKETCDMA NETWORKS

The interplay between physical layer and network layer plays an important role in designing CDMA networks. Two recent papers [4], [5] study the cross-layer admission control problem. However, onlyconstant bit rate trafficis considered in [4], [5]. Using the concept of cross-layer effective band-width proposed in this letter, we can design an optimal cross-layer CAC scheme with variable bit rate packet multimedia traffic. We use SIR outage probability as a QoS measure, which is not considered in [4], [5]. Note that, as in [5], the CAC scheme proposed in this letter is also a genuine cross-layer design in which information is exchanged between phys-ical layer and network layer in both directions. We identify the state space, decision epochs, actions, state dynamics, cost and constraints as follows.

A. State Space, Decision Epochs and Actions

The state space X of the system comprises of any state vector x= [n1, n2, . . . , nJ] ZJ+ such that the SIR outage

probability constraint can be met. Therefore, the state space of the SMDP can be defined in (8) or (9). We choose the decision epochs to be the set of all connection arrival and departure instances. Actiona(t)at decision epochtis defined asa(t) =

[a1(t), a2(t), . . . , aJ(t)], where aj(t) denotes the action for

classj connections. Ifaj(t) = 1, admit a classj connection.

Ifaj(t) = 0, the connection is rejected. The action space can

be defined as A =a:a∈ {0,1}J, j= 1,2, . . . , J. For a given state x X, a selected action should not result in a transition to a state that is not in X. The action space of a given statex∈X is defined asAx={a∈A:aj = 0if (n+

eu

j) X, j = 1,2, . . . , J anda = (0,0, . . . ,0) ifx =

(0,0, . . . ,0)}. where eu

j ∈ {0,1}J denotes a row vector

containing only zeros except for the jth component, which is 1,x+eu

j corresponds to an increase of the number of class

j users by 1. B. State Dynamics

Assume that the class j connection arrival process is Pois-son with rate λj and the service duration is exponentially

distributed with mean1/μj. The state dynamics can be

char-acterized by the state transition probabilities of the embedded chain pxy(a) and the expected sojourn time τx(a) for each

state-action pair. τx(a) = J j=1λjaj+ J j=1μjnj 1 .

The state transition probabilities of the embedded Markov chain are pxy(a) = ⎧ ⎨ ⎩ λjajτx(a), ify=x+euj, j= 1,2, . . . , J μjnjτx(a), ify=x−euj, j= 1,2, . . . , J 0, otherwise. (10) C. Performance Criterion, Cost Function, and Constraints

The average cost is considered as the performance criterion in this letter. Based on the action a taken in a state x, a cost c(x, a) occurs to the network. Authors in [4] show that the blocking probability can be expressed as an average cost

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αij(L, N, h, ξ, ω, s, t) = 1 stlogE esRij[0,tij/N , L, N∈Z+, h, ξ, ω, s, t∈IR+, (6) X = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x∈ZJ+: 1 s∗ 2π∂s22 s∗ J j=1 nj i=1 αi j exp ⎡ ⎣s∗ ⎛ ⎝J j=1 njαj−1 ⎞ ⎠ ⎤ ⎦ζ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ . (9) TABLE I

PARAMETERS USED IN NUMERICAL EXAMPLES.

Parameter Notation Value

system bandwidth W 5 MHz

spreading gain N 512

target SIR for voice traffic ω1 7 dB

estimated average channel gain for voice traffic |¯h1|2 1 channel estimation error variance for voice traffic ξ21 0.02 data transmission rate in state ON for voice traffic R1 15 kbps

rate from ON to OFF for voice traffic α1 0.4 rate from OFF to ON for voice traffic β1 0.4 service rate for voice traffic μ1 0.005

target SIR for video traffic ω2 10 dB

estimated average channel gain for video traffic |¯h2|2 1 channel estimation error variance for video traffic ξ22 0.05 data transmission rate in state NORMAL for video traffic R21 30 kbps

data transmission rate in state BURST for video traffic R22 60 kbps rate from state NORMAL to state BURST for video traffic α2 0.024 rate from state BURST to state NORMAL for video traffic β2 0.076

service rate for video traffic μ2 0.004

criterion in the CAC setting. Following [4], we define the cost as c(x, a) =

J

j=1

wj(1−aj), wherewj IR+ is the weight

associated with classj.

In the formulated problem, the SIR outage probability constraint in the system can be guaranteed by restricting the state space in (8) and (9). In addition, the blocking probability for classj can be defined as the fraction of time the system is in a set of statesXb

j ⊂X and the chosen action is in a set

of actions Axbj A, where xbj X b j and Axbj = {a A : aj = 0}, Pjb = xXjb aA xbjτx(a)/ xX aAτx(a).

The blocking probability constraints can be expressed as

Pb

j ≤γj, j= 1,2, . . . , J,which can be addressed in the linear

programming formulation by defining a cost function related to these constraintscbj(x, a) = (1−aj), j= 1,2, . . . , J.

D. Linear Programming Solution to the SMDP

The optimal policyu∗ of the SMDP is obtained by solving the following linear program [13].

min zxa0,xX,aAx x∈X a∈Ax J j=1 wj(1−aj)τx(a)zxa subject to a∈Ay zya− x∈X a∈Ax pxy(a)zxa = 0, y∈X x∈X a∈Ax zxaτx(a) = 1, x∈X a∈Ax (1−aj)zxaτx(a) γj, j= 1,2, . . . , J. (11) V. NUMERICALEXAMPLES

The numerical values used for the system parameters in the numerical examples are given in Table 1. We consider two classes of traffic, voice and MPEG video. The voice traffic is modeled as an ON/OFF process. The rate in state ON is 15 kbps corresponding to an equivalent spreading gainN = 256.

A Markov model with two states, NORMAL and BURST, is used for the MPEG traffic. The rate in state NORMAL is 30 kbps corresponding to an equivalent spreading gainN = 128

and the rate in state BURST is two times of that in state NORMAL.

We show the performance of cross-layer effective band-width by comparing the admissible regions calculated from (8) and (9) with that from the simulations. In the simulations, we keep adding more connections to the system until the SIR outage probability constraint, 0.005, is violated. Fig. 1 shows the results. We can see that the admissible regions obtained from the effective bandwidth approaches using both Chernoff bound and improved bound are smaller than that from the simulations, which means that the approaches using cross-layer effective bandwidth are conservative compared to the

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0 5 10 15 20 25 0 50 100 150 200 250 300 350 400

Number of video connections

Number of voice connections

Simulation

Cross−layer effective bandwidth (Improved bound) Cross−layer effective bandwidth (Chernoff bound)

Fig. 1. Admissible regions obtained from cross-layer effective bandwidth and simulations.

simulations, especially when Chernoff bound is used. How-ever, we observe that the admissible region using improved bound can have good approximation to the simulations. This shows the effectiveness of the concept of cross-layer effective bandwidth. We use improved bound in the following examples. We compare the admissible regions obtained from cross-layer effective bandwidth to those from peak rate, mean rate and network layer effective bandwidth allocation schemes. Since SIR outage is not allowed in [4], [5], all connections should be allocated with their peak rates to guarantee the SIR at all time instants. Therefore, [4] and [5] are examples of the peak rate scheme. In the mean rate allocation scheme, each connection is allocated with its mean rate. In the network layer effective bandwidth approach [7], only the statistical characteristics of the traffic is considered and LMMSE re-ceivers are not used. Fig. 2 shows the comparisons with peak rate and mean rate schemes. It is observed that the admissible region obtained from the peak rate scheme is much smaller than that from cross-layer effective bandwidth scheme, which means significantly lower network utilization in the peak rate scheme. The mean rate approach can increase the network utilization, but physical layer SIR outage probability cannot be guaranteed. By introducing a small SIR outage probability, 0.005, the proposed cross-layer effective bandwidth approach can increase network utilization substantially.

Fig. 3 shows the admissible regions obtained from cross-layer effective bandwidth and network cross-layer effective band-width approaches. Two values of the number of resolvable paths are used in this example, L = 1 and L = 5. We

can see that the admissible region obtained from the network layer approach is much smaller that those from cross-layer approaches. This illustrates the importance of considering the physical layer in designing CDMA networks.

VI. CONCLUSIONS

We have presented a novel concept of cross-layer effective bandwidth for variable bit rate multimedia traffic in packet CDMA networks taking into account both physical layer linear minimum mean square error (LMMSE) receivers and

0 5 10 15 20 25 30 35 0 50 100 150 200 250 300 350 400 450

Number of video connections

Number of voice connections

Mean rate

Cross−layer effective bandwidth (SIR outage probability = 0.005) Peak rate

Fig. 2. Admissible regions obtained from mean rate, cross-layer effective bandwidth and peak rate schemes.

0 10 20 30 40 50 60 70 80 0 100 200 300 400 500 600

Number of video connections

Number of voice connections

Cross−layer effective bandwidth ( L = 1) Cross−layer effective bandwidth ( L = 5) Network layer effective bandwidth

Fig. 3. Admissible regions obtained from cross-layer effective bandwidth and network layer effective bandwidth schemes.

statistical characteristics of the packet traffic. The proposed concept can be a useful tool in the design, analysis and control of packet CDMA networks. An optimal connection admission control scheme was developed based on the concept of cross-layer effective bandwidth. It was shown in numerical results that the admissible region obtained from the cross-layer effective bandwidth can have good approximation to the simulations. We also observed a very significant gain in the capacity when a small SIR outage probability is introduced in packet CDMA networks with LMMSE receivers. Further study is in progress to apply the concept of cross-layer effective bandwidth to other applications, such as tariffs and tele-traffic analysis.

REFERENCES

[1] D. N. C. Tse and S. Hanly, “Linear multiuser receivers: effective interference, effective bandwith and user capacity,”IEEE Trans. Inform. Theory, vol. 45, pp. 641–657, Mar. 1999.

[2] J. Evans and D. N. C. Tse, “Large system performance of linear multiuser receivers in multipath fading channels,”IEEE Trans. Inform. Theory, vol. 46, pp. 2059–2078, Sept. 2000.

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[3] F. Kelly, “Notes on effective bandwidths,” in Stochastic Networks: Theory and Applications (F. Kelly, S. Zachary, and I. Ziedins, eds.), pp. 141–168, Oxford Press, 1996.

[4] S. Singh, V. Krishnamurthy, and H. V. Poor, “Integrated voice/data call admission control for wireless DS-CDMA systems with fading,”IEEE Trans. Signal Proc., vol. 50, pp. 1483–1495, June 2002.

[5] C. Comaniciu and H. V. Poor, “Jointly optimal power and admission control for delay sensitive traffic in CDMA networks with LMMSE receivers,”IEEE Trans. Signal Proc., vol. 51, pp. 2031–2042, Aug. 2003. [6] F. Yu, V. Krishnamurthy and V. C. M. Leung, “Cross-layer optimal connection admission control for variable bit rate multimedia traffic in packet wireless CDMA networks,”IEEE Trans. Signal Proc., to appear, 2006.

[7] J. S. Evans and D. Everit, “Effective bandwidth-based admission control for mutliservice CDMA cellular networks,”IEEE Trans. Veh. Technol., vol. 48, pp. 36–46, Jan. 1999.

[8] J. Zhang, “Bursty traffic meets fading: a cross-layer design perspective,” inProc. of the 40th Allerton Conf. Commun., Control, and Computing, Oct. 2002.

[9] S. Verdu, “Multiuser detection,” in Advances in Statistical Signal Processing(H. Poor and J. Thomas, eds.). Greenwich, CT.: JAI Press, 1993.

[10] M. Schwartz,Broadband Integrated Networks. Prentice Hall, 1996. [11] H. Chernoff, “A measure of asymptotic efficiency for tests of a

hy-pothesis based on the sum of observation,”Ann. Math. Statist., vol. 23, pp. 493–507, 1952.

[12] R. Bahadur and R. Rao, “On deviations of the sample mean,”Ann. Math. Statist., vol. 31, pp. 1051–1027, 1960.

[13] H. Tijms, Stochastic Modeling and Analysis: a Computational Ap-proach. New York: Wiley, 1986.

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